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Mechanics

Mechanics

Although Leonardo was interested in mechanics for most of his mature life, he would appear to have turned more and more of his attention to it from 1508 on. It is difficult to construct a unified and consistent picture of his mechanics in detail, but the major trends, concepts, and influences can be delineated with some firmness.1 Statics may be considered first, since this area of theoretical mechanics greatly attracted him and his earliest influences in this field probably came from the medieval science of weights. To this he later added Archimedes’On the Equilibrium of Planes, with a consequent interest in the development of a procedure for determining the centers of gravity of sundry geometrical magnitudes.

In his usual fashion Leonardo absorbed the ideas of his predecessors, turned them in practical and experiential directions, and developed his own system of nomenclature— for example, he called the arms of balances “braccia” while in levers “lieva” is the arm of the lever to which the power is applied and “contralieva” the arm in which the resistance lies. Influenced by the Scholastics, he called the position of horizontal equilibrium the “sito dell’equalita” (or sometimes the position “equale allo orizzonte” ). Pendent weights are simply “pesi,” or “pesi attacati,” or “pesi appiccati”; the cords supporting them—or, more generally, the lines of force in which the weights or forces act or in which they are applied—are called “appendicoli.” Leonardo further distinguished between “braccia reali o linee corporee,” the actual lever arms, and “braccia potenziali o spirituali osemireali,” the potential or effective arms. The potential arm is the horizontal distance to the vertical (that is, the “linea central” ) through the center of motion in the case of bent levers, or, more generally, the perpendicular distance to the center of motion. from the line of force about the center of motion. He often called the center of motion the “centro del circunvolubile,” or simply the “polo.”

The classical law of the lever appears again and again in Leonardo’s notebooks. For example,Codex Atlanticus, folio 176v-d, states, “The ratio of the weights which hold the arms of the balance parallel to the horizon is the same as that of the arms, but is an inverse one.” In Codex Arundel, folio 1v, the law appears formulaically as W2 = (W1. s1)/s2: “Multiply the longer arm of the balance by the weight it supports and divide the product by the shorter arm, and the result will be the weight which, when placed on the shorter arm, resists the descent of the longer arm, the arms of the balance being above all basically balanced.”

A few considerations that prove the influence of the medieval science of weights upon Leonardo are in order. The medieval science of weights consisted essentially of the following corpus of works:2 Pseudo Euclid, Liber de ponderoso et levi, a geometrical treatment of basic Aristotelian ideas relating forces, volumes, weights, and velocities; Pseudo-Archimedes, De ponderibus Archimenidis (also entitled De incidentibus in humidum), essentially a work of hydrostatics; an anonymous tract from the Greek, De canonio, treating of the Roman balance or steelyard by reduction to a theoretical balance; Thabit ibn Qurra, Liber Karastonis, another treatment of the Roman balance; Jordanus de Nemore, Elementa de ponderibus (also entitled Elementa super demonstrationem ponderum), a work existing in many manuscripts and complemented by several reworked versions of the late thirteenth and the fourteenth centuries, which was marked by the first use of the concept of positional gravity (gravitas secundum situm), by a false demonstration of the law of the bent lever, and by an elegant proof of the law of the lever on the basis of the principle of virtual displacements; an anonymous Liber de ponderibus (called, by its modern editor, Version P), which contains a kind of short Peripatetic commentary to the enunciations of Jordanus; a Liber de ratione ponderis, also attributed to ordanus, which is a greatly expanded and corrected version of the Elementa in four parts and is distinguished by a more correct use of the concept of positional gravity, by a superb proof of the law of the bent lever based on the principle of virtual displacements, by the same proof of the law of the straight lever given in the Elementa, and by a remarkable and sound proof of the law of the equilibrium of connected weights on adjacent inclined planes that is also based on the principle of virtual displacements—thereby constituting the first correct statement and proof of the inclined plane law—and including a number of practical problems in statics and dynamics, of which the most noteworthy are those connected with the bent lever in part III; and Blasius of Parma, Tractatus de ponderibus, a rather inept treatment of the problems of statics and hydrostatics based on the preceding works.

The whole corpus is marked methodologically by its geometric form and conceptually by its use of dynamics (particularly the principle of virtual velocities or the principle of virtual displacements in one form or another) in application to the basic statical problems and conclusions inherited from Greek antiquity. Leonardo was much influenced by the corpus’ general dynamical approach as well as by particular conclusions of specific works. Reasonably conclusive evidence exists to show that Leonardo had read Pseudo-Archimedes, De canonio, and Thybit ibn Qurra, while completely conclusive evidence reveals his knowledge of the Elementa, De ratione ponderis, and Blasius. It is reasonable to suppose that lie also saw the other works of the corpus, since they were so often included in the same manuscripts as the works he did read.

The two works upon which Leonardo drew most heavily were the Elementa de ponderibus and the Liber de ratione ponderis. The key passage that shows decisively that Leonardo read both of them occurs in Codex Atlanticus, folios 154v-a and r-a. In this passage, whose significance has not been properly recognized before, Leonardo presents a close Italian translation of all the postulates from the Elementa (E.01-E.07) and the enunciations of the first two propositions (without proofs), together with a definition that precedes the proof of proposition E.2.3 He then shifts to the Liber de ratione ponderis and includes the enunciations of all the propositions of the first two parts (except R2.05) and the first two propositions of the third part.4 It seems likely that Leonardo made his translation from a single manuscript that contained both works and, after starting his translation of the Elementa, suddenly realized the superiority of the De ratione ponderis. He may also have translated the rest of part III, for he seems to have been influenced by propositions R3.05 and R3.06, the last two propositions in part III, and perhaps also by R3.04 in passages not considered here (see MS A, fols. Ir, 33v). Another page in Codex Atlanticus, folio 165v-a and v-c, contains all of the enunciations of the propositions of part IV (except for R4.07, R4. 11-12, and R4.16). The two passages from the Codex Atlanticus together establish that Leonardo had complete knowledge of the best and the most original work in the corpus of the medieval science of weights.

The passages cited do not, of course, show what Leonardo did with that knowledge, although others do. For example in MS G, foilo 79r, Leonardo refutes the incorrect proof of the second part of proposition states: “When the beam of a balance of equal arm lengths is in the horizontal position, if equal weights are suspended [from its extremities], it will not leave the horizontal position, and if it should be moved from the horizontal position, it will revert to it.” 5 The second part is true for a material beam supported from above, since the elevation of one of the arms removes the center of gravity from the vertical line through the fulcrum and, accordingly, the balance beam returns to the horizontal position as the center of gravity seeks the vertical. The medieval proof in E.2 and R1.02, however, treats the balance as if it were a theoretical balance (with weightless beam) and attempts to show that it would return to the horizontal position. Based on the false use of the concept of positional gravity, it asserts that the weight above the line of horizontal equilibrium has a greater gravity according to its position than the weight depressed below the line, for if both weights tended to move downward, the arc on which the upper weight would move intercepts more of the vertical than the arc on which the lower weight would tent to move. Leonardo’s refutation is based on showing that because the weights are connected, the actual arcs to be compared are oppositely directed and so have equal obliquities—and thus the superior weight enjoys no positional advantage. In MS E, folio 59r, Leonardo seems also to give the correct explanation for the return to horizontal equilibrium of the material beam.

A further response to the Liber de ratione ponderis is in Codex Atlanticus, folio 654v-c, where Leonardo again translates proposition R1.09 and paraphrases proposition R1.10 of the medieval work: “The equality of the declination conserves the equality of the weights. If the ratios of the weights and the obliquities on which they are placed are the same but inverse, the weights will remain equal in gravity and motion” ( “La equalita della declinazione osserva la equalita de’ pesi. Se Ie proporzionic de’ pesi e dell’ obbliqua dove si posano, saranno equali, ma converse, essi pesi resteranno equali in gravita e in moto” ). The equivalent propositions from the Liber de ratione ponderis were “R1.09. Equalitas declinationis identitatem conservat ponderis” and “R1.10. Si per diversarum obliquitatum vias duo pondera descendant, fueritque declinationum et ponderum una proportio eodum ordine sumpta, una erit utriusque virtus in descendendo.” 6 While Leonardo preserve R1.09 exactly in holding that positional weight on the incline is everywhere the same as long as the incline’s declination is the same, his rephrasing of R1.10 indicates that he adopted a different measure of “declination.”

R. Marcolongo believed that Leonardo measured obliquity by the ratio of the common altitude of the inclines to the length of the incline (that is, by the sine of the nagle of inclination), while Jordanus had measured obliquity by the length of the incline that intercepts the common altitude (that is, by the cosecant of hte angle).7 Thus, ifp1 and p2 are connected weights placed on the inclined planes that are of lengths l1l2 and l2, respectively, with common altitude h, then, according to Marcolongo’s view of Leonardo’s method, p1/p2 = (h/l2)/(h/l1), and thus p1/p2 = l1/l2, as Jordanus held. Hence, if Marcolongo is correct, Leonardo had absorbed the correct exposition of the inclined plane problem from the Liber de ratione ponderis. Perhaps he did, but it is not exhibited in this passage, for the figure accompanying the passage (Figure 1), with its numerical designations of 2 and 1

on the bases cut off by the vertical, shows that Leonardo believed the weights on the inclines to be inversely proportional to the tangents of the angles of inclination rather than to the sines—and such a solution is clearly incorrect. The same incorrect solution is apparent in MS G, folio 77v, where again equilibrium of the two weights is preserved when the weights are in the same ratio as the bases cut off by the common altitude (thus implying that the weights are inversely proportional to the tangents).

Other passages give evidence of Leonardo’s vacillating methods and confusion. One (Codex on Flight, fol. 4r) consists of two paragraphs that apply to the same figure (Figure 2). The first is evidently an explanation of a proposition expressed elsewhere (MS E, fol. 75r) to the effect that although equal weights balance each other on the equal arms of a balance, they do not do so if they are put on inclines of different obliquity. It states:

The weight q, because of the right angle n [perpendicularly] above point e in line df, weighs 2/3 of its natural weight, which was 3 pounds, and so has a residual force [“che resta in potenzia’] [along nq] of 2 pounds; and the weight p, whose natural weight was also 3 pounds, has a residual force of 1 pound [along nip] because of right angle m [perpendicularly] above point g in line hd. [Therefore, p and q are not in equilibrium on these inclines.]

The bracketed material has been added as clearly implicit, and so far this analysis seems to be entirely correct. It is evident from the figure that Leonardo has applied the concept of potential lever arm (implying static moment) to the determination of the component of weight along the incline, so that F1. dn = W1. de, where F1 is the component of the natural weight W1 along he incline, dn is the potential lever arm through which F1 acts around fulcrum d, and de is the lever

arm through which W1would act when hanging from n. Hence,F1 = W1. (de/dn) = W1. sin ∠dne. But ∠dne = ∠α, and so we have the correct formulation F1 = W1. sin α. In the same way for weight p, it can be shown that F2 = W2. sin β. And since Leonardo apparently constructed de = 2df/3, dg==hd/3, and W1 = W2 = 3, obviously F1 = 2 and F2 = 1 and the weights are not in equilibrium.

In the second paragraph, which also pertains to the figure, he changes p and i, each of which was initially equal to three pounds, to two pounds, and one pound, respectively, with the object of determining whether the adjusted weights would be in equilibrium:

So now we have one pound against two pounds. And because the obliquities da and de on which these weights are placed are not in the same ratio as the weights, that is 2 to 1 [the weights are not in equilibrium]; like the said weights [in the first paragraph?], they alter natural gravities [but they are not in equilibrium] because the obliquity da exceeds the obliquity dc, or contains the obliquity dc, 21/2 times, as is demonstrated by their bases ab and bc, whose ratio is a double sesquialternate ratio [that is, 5:2], while the ratio of the weights will be a double ratio [that is, 2:1].

It is abundantly clear, at least in the second paragraph, that Leonardo was assuming that for equilibrium the weights ought to be directly proportional to the bases (and thus inversely proportional to the tangents); and since the weights are not as the base lines, they are not in equilibrium. In fact, in adding line d[x] Leonardo was indicating the declination he thought would establish the equilibrium of p and q, weights of two pounds and one pound, respectively, for it is obvious that the horizontal distances b[x] and be are also related as 2:1. this is further confirmation that Leonardo measured declination by the tangent.

Assuming that both paragraphs were written at the same time, it is apparent that Leonardo then thought that both methods of determining the effective weight on an incline—the method using the concept of the lever and the technique of using obliquities measured

by tangents—were correct. A similar confusion is apparent in his treatment of the tensions in strings. But there is still another figure (Figure 3), which has accompanying it in MS H, folio 81(33)v, the following brief statement: “On the balance, weight ab will be as weight cd.” If Leonardo was assuming that the weights are of the same material with equal thickness and, as it seems in the figure, that they are of the same width, with their lengths equal to the lengths of the vertical and the incline, respectively, thus producing weights that are proportional to these lengths, than he was indeed giving a correct example of the inclinedplane principle that may well reflect proposition R1.10 of the Liber de ratione ponderis. There is one further solution of the inclined-plane problem, on MS A, folio 21v, that is totally erroneous and, so far as is known, unique. It is not discussed here; the reader is referred to Duhem’s treatment, with the caution that Duhem’s conclusion that it was derived from pappus’ erroneous solution is questionable,8

As important as Leonardo’s responses to proposition R1.10 are his responses to the bent-lever proposition, R1.08, of the Liber de ratione ponderis. In Codex Arundel, folio 32v, he presents another Italian translation of it 9in addition to the translation already noted in his omnibus collection of translations of the enunciations of the medieval work). In this new translation he writes of the bent-lever law as “tested” : “Tested, If the arms of the balance are unequal and thier juncture in the fulcrum is angular, and if their termini are equally distant from the central [that is, vertical] line through the fulcrum, with equal weights applied there [at the termini], they will weigh equally [that is, be in equilibrium]” —or “Sperimentata. Se le braccia della bilancia fieno ineuali, e la lor congiunzione nel polsa sia angulare, se i termini lor fieno equalmente distanti alla linea central del polo, li pesi appiccativi, essendo equali, equalmente peseranno,” a trnaslation of the Latin text, “R1.08. Si inequlia fuerint brachia libre, et in centro motus angulum fecerint, si termini eorum ad directionem hinc inde equaliter accesserint, equlia appensa, in hac dispositione equaliter ponderbunt,” 9 Other, somewhat confused passages indicate that l;eonardo had indeed absorbed the significance of hte passage he had twice translated.10

More important, Leonardo extended the bent-lever law beyond the special case of equal weights at equal horizontal distances, given in R1.08, to cases in which the more general law of the bent lever—“weights are inversely proportional to the horizontal distances’—is applied. He did this in large number of problems to which he applied his concept of “potential lever arm.” The first passage notable for this concept is in

MS E, folio 72v, and probably derives from proposition R3.05 of the Liber de ratione ponderis.11 Here Leonardo wrote (see Figure 4): “The ratio that space mn has to space nb is the same as the ratio that the weight which has descended to d has to the weight it had in position b. It follows that, mn being 9/11 of nb, the weight in d is 9/10 (!9/11) of the weight it had in height b.” In this passenger n is the center of motion and mn is the potential lever arm of the weight in position d. His use of the potential lever arm is also illustrated in

a passage that can be reconstructed from MS E, folio 65r, as follows (Figure 5). A bar at is pivoted at a; a weight o is suspended from t at m and a second weight acts on t in a direction tn perpendicular to that of the first weight. The weights at mand n necessary to keep the bar in equilibrium are to be determined. In this determination Leonardo took ab and ac as the potential lever arms (and so labeled them), so that the weights m and n are inversely proportional to the potential lever arms ab and ac. The same kind of problem is illustrated in a passage in Codex Atlanticus, folio 268v-b, which can be summarized by reference to Figure 6. Cord ab supports a weight n, which is

pulled to the position indicated by a tangential force along nf that is given a value of 1 and there keeps an in equilibrium. The passage indicates that weight 1, acting through, a distance of one unit (the potential lever arm an.), keeps in equilibrium a weight n of 4, acting through a distance of one unit (the potential counterlever ac). A similar use of “potential lever arm” is found in problems like that illustrated in Figure 7, taken from MS M, folio 40r. Here the potential lever arm is an, the perpendicular drawn from the line of force fP to the center of motion a. Hence the weights suspended from p and m are related inversely as the distances an and am. These and similar problems reveal Leonardo’s acute awareness of the proper factors of horizontal distance and force determining the static moment about a point.

The concept of potential lever arm also played a crucial role in Leonardo’s effort to analyze the tension

in strings. Before examining that role, it must be noted that Leonardo often used an incorrect rule based on tangents rather than sines, a rule similar to that which he mistakenly applied to the problem of the inclined plane. An example of the incorrect procedure appears in MS E , folio 66v (see Figure 8):

The heavy body suspended in the angle of the cord divides the weight to these cords in the ratio of the angles included between the said cords and the central [that is, vertical] line of the weight. Proof. Let the angle of the said cord be bac, in which is suspended heavy body g by cord ag. Then let this angle be cut in the position of equality [that is, in the horizontal direction] by lines fb. Then draw the perpendicular da to angle a

and it will be in the direct continuation of cord ag, and the ratio which space df has to space db, the weight [that is, tension] in cord ba will have to the weight [that is, tension] in cord fa.

Thus, with da the common altitude and df and db used as the measures of the angles at a, Leonardo is actually measuring the tensions by the inverse ratio of the tangents. This same incorrect procedure appears many times in the notebooks (for instance, MS E, fols. 67v, 68r, 68v, 69r, 69v, 71r; Codex Arundel, fol. 117v; MS G, fol. 39v).

In addition to this faulty method Leonardo in some instances employed a correct procedure based on the concept of the potential lever arm. In Codex Arundel, folio 1v, Leonardo wrote that “the weight 3 is not distributed to the real arms of the balance in the same [although inverse] ratio of these arms but in the [inverse] ratio of the potential arms” (see Figure 9.)

The cord is attached to a horizontal beam and a weight is hung from the cord. In the figure ab and ac are the potential arms. This equivalent to a theorem that could be expressed in modern terms as “the moments of two concurrent forces around a point on the resultant are mutually equal.” Leonardo’s theorem, however, does not allow the calculation of the actual tensions in the strings. By locating the center of the moments first on one and then on the other of the concurrent forces, however, Leonardo discovered how to find the tensions in the segments of a string supporting a weight. In Codex Arundel, folio 6r (see Figure 10), he stated:

Here the potential lever db is six times the potential counterlever be. Whence it follows that one pound placed in the force line “appendiculo” dn is equal in power to six pounds in the semireal force line

ca, and to another six pounds of power conjoined at b. Therefore, the cord aeb by means of one pound placed in line dn has the [total] effect of twelve pounds.

All of which is to say that Leonardo has determined that there are six pounds of tension in each segment of the cord. The general procedure is exhibited in MS E, folio 65r (see Figure 11). Under the figure is the caption “a is the pole [that is, fulcrum] of the angular balance [with arms] ad and af, and their force lines [’appendiculi’] are dnand fc.” This applies to the figure on the left and indicates that of and da are the lever arms on which the tension in cb and the weight hanging at b act. Leonardo then went on to say: “The greater the angle of the cord which supports weight n in the middle of the cord, the smaller becomes the potential arm [that is, ac in the figure on the right] and the greater becomes the potential counterarm [that is, ba] supporting the weight.” Since Leonardo has drawn the figure so that ab is four times ac and has marked the weight of n as l, the obvious implication is that the tension in cord df will be 4.

In these last examples the weight hangs from the center of the cord. But in Codex Arundel, folio 6v, the same analysis is applied to a weight suspended at a point other than the middle of the cord (Figure 12).

Here the weight n is supported by two different forces, mf and mb. Now it is necessary to find the potential levers and counterlevers of these two forces bm and fm. For the force [at] b [with f the fulcrum of the potential lever] the [potential] lever arm is fe and arm fe and the [potential] counterlever is fa. Thus for the lever arm fe the force line [ “appendiculo” ] is eb along which the motor b is applied; and for the counterlever fa the force line is an, which supports weight n. Having arranged the balance of the power and resistance of motor and weight, it is necessary to see what ratio lever fe. has to counterlever fa; which lever fe is 21/22 of counterlever fa. Therefore b suffers 22 [pounds of

tension] when n is 21. In the second disposition [with b instead off as the fulcrum of a potential lever],bc is the [potential] lever arm ba the [potential] counter-lever. For be the force line is cf along which the motor f is applied and weight n is applied along force line an. Now it is necessary to see what ratio lever bc has to counterlever ba, which counterlever is 1/3 of the lever. Therefore, one pound of force in f resists three pounds of weight in ba; and 21/22 of the three pounds in n, when placed at b, resist twenty-two placed in ba…. Thus is completed the rule for calculating the unequal arms of the angular cord.

In a similar problem in Codex Arundel, folio 4v, Leonardo determined the tensions of the string segments; in this case the strings are no longer attached or fixed to a beam but are suspended from two pulley wheels from which also hang two equal weights (Figure 13). Here abc is apparently an equilateral

triangle, cd is the potential arm for the tension in string ba acting about fulcrum c and ec is the potential counterlever through which the weight 2 at a acts. If an equilateral triangle was intended, then, by Leonardo’s procedure, the tension in each segment of the string ought to be , rather than 1.5, as Leonardo miscalculated it. It is clear from all of these examples in which the tension is calculated that Leonardo was using a theorem that he understood as: “The ratio of the tension in a cord segment to the weight supported by the cord is equal to the inverse ratio of the potential lever arms through which the tension and weight act, where the fulcrum of the potential bent lever is in the point of support of the other segment of the cord.” This is equivalent to a theorem in the composition of moments: “If one considers two concurrent forces and their resultant the moment of the resultant about a point taken on one of the two concurrent forces is equal to the moment of the other concurrent force about the same point.” The discovery and use of this basic concept in analyzing string tensions was Leonardo’s most original development in statics beyond the medieval science of weights that he had inherited. Unfortunately, like most of Leonardo’s investigations, it exerted no influence on those of his successors.

Another area of statics in which the medieval science of weights may have influenced Leonardo was that in which a determination is made of the partial forces in the supports of a beam where the beam itself supports a weight. Proposition R3.06 in the Liber de ratione ponderis states: “A weight not suspended in the middle [of a beam] makes the shorter part heavier according to the ratio of the longer part to the shorter part.” 12 The proof indicates that the partial forces in the supports are inversely related to the distances from the principal weight to the supports. In Codex Arundel, folio 8v (see Figure 14), Leonardo arrived at a similar conclusion:

The beam which is suspended from its extremities by two cords of equal height divides its weight equally in each cord. If the beam is suspended by its extremities at an equal height, and in its midpoint a weight is hung, then the gravity of such a weight is equally distributed to the supports of the beam. But the weight which is moved from the middle of the beam toward one of its extremities becomes lighter at the extremity away from which it was moved, or heavier at the other extremity, by a weight which has the same ration to the total weight as the motion completed by the weight [that is,its distance moved from the center] has to the whole beam.

Leonardo also absorbed the concept, so prevalent in the medieval science of weights (particularly in theDe canonio, in Thabit ibn Qurra’s Liber karastonis, and in part II of the Liber de ratione ponderis), that a

segment of a solid beam may be replaced by a weight hung from the midpoint of a weightless arm of the same length and position. For example, see Leonardo’s exposition in MS A, folio 5r (see Figure 15):

If a balance has a weight which is similar [that is, equal] in length to one of its arms, the weight being inn of six pounds, how many pounds are to be placed in f to resist it [that is, which will be in equilibrium with it]? I say that three pounds will suffice, for if weight mn is as long as one of its arms, you could judge that it may be replaced in the middle of the balance arm at point a; therefore, if six pounds are in a, another six pounds placed at r would produce resistance to them [that is, be in equilibrium with them], and, if you proceed as before in point r [but now] in the extremity of the balance, three pounds will produce the [necessary] resistance to them.

This replacement doctrine was the key step in solving the problem of the Roman balance in all of the above-noted tracts.13

The medieval science of weights was not the only influence upon Leonardo’s statics, however, since it may be documented that he also, perhaps at a later time, read book I of Archimedes’ On the Equilibrium of Planes. As a result he seems to have begun a work on centers of gravity in about 1508, as may be seen in a series of passages in Codex Arundel. Since these passages have already been translated and analyzed in rather complete detail elsewhere,14 only their content and objectives will be given here. Preliminary to Leonardo’s propositions on centers of gravity are a number of passages distinguishing three centers of a figure that has weight (see Codex Arundel, fol. 72v): “The first is the center of its natural gravity, the second [the center] of its accidental gravity, and the third is [the center] of the magnitude of this body.” On folio 123v of this manuscript the centers are defined:

The center of the magnitude of bodies is placed in the middle with respect to the length, breadth and thickness of these bodies. The center of the accidental gravity of these bodies is placed in the middle with respect to the parts which resist one another by standing in equilibrium. The center of natural gravity is that which divides a body into two parts equal in weight and quantity.

It is clear from many other passages that the center of natural gravity is the symmetrical center with respect to weight. Hence, the center of natural gravity of a beam lies in its center, and that center would not be disturbed by hanging equal weights on its extremities. If unequal weights are applied, however, there is a shift of the center of gravity, which is now called the center of accidental gravity, the weights having assumed accidental gravities by their positions on the unequal arms. The doctrine of the three centers can be traced to Scholastic writings of Nicole Oresme, Albert of Saxony, Marsilius of Inghen (and, no doubt, others).15 In all of these preliminary considerations Leonardo assumed that a body or a system of bodies is in equilibrium when supported from its center of gravity (be it natural or accidental). It should also be observed that Leonardo assumed the law of the lever as being proved before setting out to prove his Archimedean-like propositions.

The first Archimedean passage to note is in Codex Arundel, folio 16v, where Leonardo includes a series of statements on equilibrium that is drawn in significant part from the postulates and early propositions of book I of On the Equilibrium of Planes. His terminology suggests that when Leonardo wrote this passage, he was using the translation of Jacobus Cremonensis (ca. 1450).16

More important than this passage are the propositions and proofs on centers of gravity, framed under the influence of Archimedes’ work, which Leonardo specifically cites in a number of instances. The order for these propositions that Leonardo’s own numeration seems to suggest is the following:

1. Codex Arundel, folio 16v, “Every triangle has the center of its gravity in the intersection of the lines which start from the angles and terminate in the centers of the sides opposite them.” This is proposition 14 of book I ofOn the Equilibrium of Planes, and Leonardo included in his “proof” an additional proof of sorts for Archimedes’ proposition 13 (since that proposition is fundamental for the proof of proposition 14), although the proof for proposition 13 ignores Archimedes’ superb geometrical demonstration. Depending, as it does, on balance considerations, Leonardo’s proof is more like Archimedes’ second proof of proposition 13. Leonardo’s proof of proposition 14 is for an equilateral triangle, but at its end he notes that it applies as well to scalene triangles.

2. Codex Arundel, folio 16r: “The center of gravity of any two equal triangles lies in the middle of the line beginning at the center of one triangle and terminating in the center of gravity of the other triangle.” This is equivalent to proposition 4 of On the Equilibrium of Planes, but Leonardo’s proof differs from Archimedes’ in that it merely shows that the center of gravity is in the middle of the line because the weights would be in equilibrium about that point. Archimedes’ work is here cited (under the inaccurate title of De ponderibus, since the Pseudo-Archimedean work of that title was concerned with hydrostatics rather than statics).

3.Codex Arundel,folio 16r: “If two unequal triangles are in equilibrium at unequal distances, the greater will be placed at the lesser distance and the lesser at the greater distance.” This is similar to proposition 3 of Archimedes’ work. Leonardo, however, simply employed the law of the lever in his proof, which Archimedes did not do, since he did not offer a proof of the law until propositions 6 and 7.

4. Codex Arundel, folio 17v: “The center of gravity of every square of parallel sides and equal angles is equally distant from its angles.” This is a special case of proposition 10 of On the Equilibrium of Planes, but Leonardo’s proof, based once more on a balancing procedure, is not directly related to either of the proofs provided by Archimedes.

5.Codex Arundel, folio 17v: “The center of every corbel-like figure [that is, isosceles pezium] lies in the line which divides it into two equal parts when two of its sides are parallel.” This is similar to Archimedes’ proposition 15, which treated more generally of any trapezium. Leonardo made a numerical determination of where the center of gravity lies on the bisector. Although his proof is not close to Archimedes’, like Archimedes he used the law of the lever in his proof.

6.Codex Arundel, folio 17r: “The center of gravity of every equilateral pentagon is in the center of the circle which circumscribes it.” This has no equivalent in Archimedes’ work. The proof proceeds by dividing the pentagon into triangles that are shown to balance about the center of the circle. Again Leonardo used the law of the lever in his proof, much as he had in his previous propositions. The proof is immediately followed by the determination of the center of gravity of a pentagon that is not equilateral, in which the same balancing techniques are again employed. Leonardo here cited Archimedes as the authority for the law of the lever, and the designation of Archimedes’ proposition as the “fifth” is perhaps an indication that he was using William of Moerbeke’s medieval translation of On the Equilibrium of Planes instead of that of Jacobus Cremonensis, in which the equivalent proposition is number 7. Although these exhaust those propositions of Leonardo’s that are directly related to On the Equilibrium of Planes, it should be noted that Leonardo used the same balancing techniques in his effort to determine the center of gravity of a semicircle (see Codex Arundel, fols. 215r-v).

It should also be noted that Leonardo went beyond Archimedes’ treatise in one major respect-the determination of centers of gravity of solids, a subject taken up in more detail later in the century by Francesco Maurolico and Federico Commandino. Two of the propositions investigated by Leonardo may be presented here to illustrate his procedures. Both propositions concern the center of gravity of a pyramid and appear to be discoveries of Leonardo’s. the first is that the center of gravity of a pyramid (actually, a regular tetrahedron) is, at the intersection of the axes, a distance on each axis of 1/4 of its length, starting from the center of one of the faces. (By “axis” Leonardo understood a line drawn from a vertex to the center of the opposite face.) In one place (Codex Arundel, flo. 218v) he wrote of the intersection of the pyramidal axes as follows: “The inferior [interior?] axes of pyramids which arise from [a point lying at] 1/3 of the axis of their bases [that is, faces] will intersect in [a point lying at] 1/4 of their length [starting] at the base.”

Despite the confusion of singulars and plurals as well as of the expression “inferior axes,” the proposition

is clear enough, particularly since Leonardo provided both the drawing shown in Figure 16 and its explanation and, in addition, the intersection of the pyramidal axes is definitely specified as the center of gravity in another passage (fol. 193v): “The center of gravity of the [pyramidal] body of four triangular bases [that is, faces] is located at the intersection of its axes and it will be in the 1/4 part of their length.” The proof of this is actually given on the page of the original quotation about the intersection of the axes (fol. 218v), but only as a proof following a more general statement about pyramids and cones:

The center of gravity of any pyramid—round, triangular, square, or [whose base is] of any number of sides—is in the fourth part of its axis near the base. Let the pyramid be abcd with base bcd and apex a, Find the center of the base bcd, which you let be f, then find the center of face abc, which will be e, as was proved by the first [proposition]. Now draw line af, in which the center of gravity of the pyramid lies because f is the center of base bcd and the apex a is perpendicularly above f and the angles b, c and d are equally distant from fand [thus] weigh equally so that the center of gravity lies in line af. Now draw a line from angle d to the center e of face abc, cutting af in point g. I say for the aforesaid reason that the center of gravity is in line de. So, since the center is in each [line] and there can be only one center, it necessarily lies in the intersection of these lines, namely in point g, because the angles a, b, c and d are equally distant from this g.

As noted above, the proof is given only for a regular tetrahedron, none being given for the cone or for other pyramids designated in the general enunciation. It represents a rather intuitive mechanical approach, for Leonardo abruptly stated that because the angles b, c, and d weigh equally about f, the center of gravity of the base triangle, and a is perpendicularly above f, the center of gravity of the whole pyramid must lie in line af. This is reminiscent of Hero’s demonstration of the equilibrium of a triangle supported at its center with equal weights at the angles. This kind of reasoning, then, seems to be extended to the whole pyramid by Leonardo at the end of the proof in which he declared that each of the four angles is equidistant from g and, presumably, equal weights at the angles would therefore be in equilibrium if the pyramid were supported in g. It is worth noting that a generation later Maurolico gave a very neat demonstration of just such a determination of the center of gravity of a tetrahedron by the hanging of equal weights at the angles.17 Finally, one additional theorem (without proof), concerning the center of gravity of a tetrahedron, appears to have been Leonardo’s own discovery (Codex Arundel, fol. 123v):

The pyramid with triangular base has the center of its natural gravity in the [line] segment which extends from the middle of the base [that is, the midpoint of one edge] to the middle of the side [that is, edge] opposite the base; and it [the center of gravity] is located on the segment equally distant [from the termini] of the [said] line joining the base with the aforesaid side.

Despite Leonardo’s unusual and imprecise language (an attempt has been made to rectify it by bracketed additions), it is clear that he has here expressed a neat theorem to the effect that the center of gravity of the tetrahedron lies at the intersection of the segments joining the midpoint of each edge with the midpoint of the opposite edge and that each of these segments is bisected by the center of gravity. Again, it is possible that Leonardo arrived at this proposition by considering four equal weights hung at the angles. At any rate the balance procedure, whose refinements he learned from Archimedes, no doubt played some part in his discovery, however it was made.

Leonardo gave considerable attention to one other area of statics, pulley problems. Since this work perhaps belongs more to his study of machines, except for a brief discussion in the section on dynamics, the reader is referred to Marcolongo’s brief but excellent account.18

Turning to Leonardo’s knowledge of hydrostatics, it should first be noted that certain fragments from William of Moerbeke’s translation of Archimedes’ On Floating Bodies appear in the Codex Atlanticus, folios 153v-e, 153r-b, and 153r-c.19 These fragments (which occupy a single sheet bound into the codex) are not in Leonardo’s customary mirror script but appear in normal writing, from left to right. Although sometimes considered by earlier authors to have been written by Leonardo, they are now generally believed to be by some other hand.20 Whether the sheet was once the property of Leonardo or whether it was added to Leonardo’s material after his death cannot be determined with certainty—at any rate, the fragments can be identified as being from proposition 10 of book II of the Archimedean work. Whatever Leonardo’s relationship to these fragments, his notebooks reveal that he had only a sketchy and indirect knowledge of Archimedean hydrostatics, which he seems to have drawn from the medieval tradition of De ponderibus Archimenidis. Numerous passages in the notebooks reveal a general knowledge of density and specific weight (for instance, MS C, fol. 26v, MS F, fol. 70r; MS E, fol. 74v). Similarly, Leonardo certainly knew that bodies weigh less in water than in air (see MS F, fol. 69r), and in one passage (Codex Atlanticus, fol. 284v) he proposed to measure the relative resistance of water as compared with air by plunging the weight on one arm of a balance held in areial equilibrium into water and them determining how much extra weight must be added to the weight in the water to maintain the balance in equilibrium. See also MS A, folio 30v: “The weight in air exhibits the truth of its weight, the weight in water will appear to be less weight by the amount the water is heavier than the air.”

So far as is known, however, the principle of Archimedes as embraced by proposition 7 of book I of the genuineOn Floating Bodies was not precisely stated by Leonardo. Even if he did know the principle, as some have suggested, he probably would have learned it from proposition 1 of the medical De ponderibus Archimenidis. As a matter of fact, Leonardo many times repeated the first postulate of the medieval work, that bodies or elements do not have weight amid their own kind—or, as Leonardo put it (Codex Atlantics, fol. 365r-a; Codex Arundel, fol. (189r), “No part of an element weights in its element” (cf. Codex Arundel, fol. 160r). Still, he could have gotten the postulate from Blasius of Parama’s De ponderibus, a work that Leonardo knew and criticized (see MS Ashburnham 2038, fol. 2v). It is possible that since Leonardo knew the basic principle of floating bodies-that a floating body displaces its weight of liquid (see Codex Forster II2, fol. 65v)—he may have gotten it directly from proposition 5 of book I of On Floating Bodies. But even this principle appeared in one manuscript of the medieval De ponderibus Archimenidis and was incorporated into John of Murs’ version of that work, which appeared as part of his widely read Quadripartitum numerorum of 1343.21 So, then, all of the meager reflections of Archimedean hydrostatics found in Leonardo’s notebooks could easily have been drawn from medieval sources, and (with the possible exception of the disputed fragments noted above) nothing from the brilliant treatments in book II of the genuine On Floating Bodies is to be found in the great artist’s notebooks. It is worth remarking, however, that although Leonardo showed little knowledge of Archimedean hydrostatics he had considerable success in the practical hydrostatics question that arose from his study of pumps and other hydrostatic devices.22

The problems Leonardo considered concerning the hydrostatic equilibrium of liquids in communicating vessels were of two kinds: those in which the liquid is under the influence of gravity alone and those in which the liquid is under the external pressure of a piston in one of the communicating vessels. In connection with problems of the first kind, he expressly and correctly stated the law of communicating vessels in Codex Atlanticus, folio 219v-a: “The surfaces of all liquids at rest, which are joined together below, are always of equal height.” Leonardo further noted in various ways that a quantity of water will never lift another quantity of water, even if the second quantity is in a narrower vessel, to a level that is higher than its own, whatever the ratio between the surfaces of the two communicating vessels (Codex Atlanticus, fol. 165v)—although he never gave, as far as can be seen, the correct explanation: the equality of the air pressure on both surfaces of the water in the communicating vessels. In some but not all passages (see Codex Leicester, fol. 25r; Codex Atlanticus, fol. 321v-a) Leonardo did free himself from the misapplication to hydrostatic equilibrium of the principle of the equilibrium of a balance of equal arms bearing equal weights (Codex Atlanticus, fol. 206r-a; Codex Arundel, fol. 264r). His own explanations re not happy ones, fol. 264r). His own explanations are not happy ones, however. He also correctly analyzed the varying levels that would result if, to a liquid in a U-tube, were added a specifically lighter liquid which does not mix with the initial liquid (see Figure 17). In MS E, folio

74v, he indicated that if the specific weight of the initial liquid were double that of the liquid added, the free surface of the heavier liquid in limb B would be at a level halfway between the level of the free surface of the lighter liquid and the surface of contact of the two liquids (the last two surfaces being in limb A).

In problems of the second kind, in which the force of a piston is applied to the surface of the liquid in one of the communicating vessels (Figure 18), despite some passages in which he gave an incorrect or only partially true account (see MS A, fol. 45r; Codex Atlanticus, fol. 384v-a), Leonardo did compose an entirely correct and generally expressed applicable statement or rule (Codex Leicester, fol. 11r). Assuming the tube on the right to be vertical and cylindrical, Leonardo observed that the ratio between the pressing weight of the piston and the weight of water in the tube on the right, above the upper level of the water in the vessel on the left, is equal to the ratio between the area under pressure from the piston and the area

of the tube on the right (see Codex Atlanticus, fols. 20r. 206r-a, 306v-c; Codex Leicester, fol. 26r). And indeed, in a long explanation accompanying the rule in Codex Leicester, folio 11r, Leonardo approached, although still in a confused manner, the concept of pressure itself (a concept that appears in no other of his hydrostatic passages, so far as is known) and that of its uniform transmission through a liquid. Leonardo did not, however, generalize his observations to produce Pascal’s law, that any additional pressure applied to a confined liquid at its boundary will be transmitted equally to every point in the liquid. Incidentally, the above-noted passage in Codex Leicester, folio IIr, also seems to imply a significant consequence for problems of the first kind: that the pressure in an enclosed liquid increases with its depth below its highest point.

Any discussion of Leonardo’s knowledge of hydrostatics should be complemented with a few remarks regarding his observations on fluid motion. From the various quotations judiciously evaluated by Truesdell, one can single out some in which Leonardo appears as the first to express special cases of two basic laws of fluid mechanics. The first is the principle of continuity, which declares that the speed of steady flow varies inversely as the cross-sectional area of the channel. Leonardo expressed this in a number of passages, for example, in MS A, folio 57v, where he stated: “Every movement of water of equal breadth and surface will run that much faster in one place than in another, as [the water] may be less deep in the former place than in the latter” (compare MS H, fol. 54(6)v; Codex Atlanticus, fols. 80r-b: 81v-a ; and Codex Leicester, fols. 24r and, particularly, 6v). He clearly recognized the principle as implying steady discharge in Codex Atlanticus, folio 287r-b: “If the water is not added to or taken away from the river, it will pass with equal quantities in every degree of its breadth [length?], with diverse speeds and slownesses, through the various straitnesses and breadths of its length.”

The second principle of fluid motion enunciated by Leonardo was that of equal circulation. In its modern form, when applied to vortex motion, it holds that the product of speed and length is the same on each circle of flow. Leonardo expressed it, in Codex Atlanticus, folio 296v-b, as:

The helical or rather rotary motion of every liquid is so much the swifter as it is nearer to the center of its revolution. This that we set forth is a case worthy of admiration ; for the motion of the circular wheel is so much the slower as it is nearer to the center of the rotating thing. But in this case [of water] we have the same motion, through speed and length, in each whole revolution of the water, just the same in the circumference of the greatest circle as in the least ….

It could well be that Leonardo’s reference to the motion of the circular wheel was suggested by either a statement in Pseudo-Aristotle’s Mechanica (848A)—that on a rotating radius “the point which is farther from the fixed center is the quicker"—or by the first postulate of the thirteenth-century Liber de motu of Gerard of Brussels, which held that “those which are farther from the center or immobile axis are moved more [quickly]. Those which are less far are moved less [quickly].’23 At any rate, it is worthy of note that Leonardo’s statement of the principle of equal circulation is embroidered by an unsound theoretical explanation, but even so, one must agree with Truesdell’s conclusion (p. 79)a; “If Leonardo discovered these two principles from observation, he stands among the founders of western mechanics.”

The analysis can now be completed by turning to Leonardo’s more general efforts in the dynamics and kinematics of moving bodies, including those that descend under the influence of gravity. In dynamics Leonardo often expressed views that were Aristotelian or Aristotelian as modified by Scholastic writers. His noted contain a virtual flood of definitions of gravity, weight, force, motion,impetus, and percussion:

1. MS B, folio 63r: “Gravity, force, material motion and percussion are four accidental powers with which all the evident works of mortal men have their causes and their deaths.”

2. Codex Arundel, folio 37r: “Gravity is an invisible power which is created by accidental motion and infused into bodies which are removed from their natural place.”

3. Codex Atlanticus, folio 246r-a: “The power of every gravity is extended toward the center of the world.”

4. Codex Arundel, folio 37v: “Gravity, force and percussion are of such nature that each by itself alone can arise from each of the others and also each can give birth to them. And all together, and each by itself, can create motion and arise from it” and “Weight desires [to act in] a single line [that is, toward the center of the world] and force an infinitude [of lines]. Weight is of equal power throughout its life and force always weakens [as it acts]. Weight passes by nature into all its supports and exists throughout the length of these supports and completely through all their parts.”

5. Codex Atlanticus, folio 253r-c: “Force is a spiritual essence which by accidental violence is conjoined in heavy bodies deprived of their natural desires; in such bodies, it [that is, force], although of short duration, often appears [to be] of marvelous power. Force is a power that is spiritual, incorporeal, impalpable, which force is effected for a short life in bodies which by accidental violence stand outside of their natural repose. ’Spiritual,’ I say, because in it there is invisible life; ’incorporeal’ and impalpable,’ because the body in which it arises does not increase in form or in weight.”

6. MS A, folio 34v: “Force, I say to be a spiritual virtue, an invisible power, which through accidental, external violence is caused by motion and is placed and infused into bodies which are withdrawn and turned from their natural use….”

On the same page as the last Leonardo indicated that force has three “offices” ("ofizi” ) embracing an “infinitude of examples” of each. These are “drawing” ( “tirare” ), “Pushing” ( “spignere” ), and “stopping” ( “fermare” ). Force arises in two ways: by the rapid expansion of a rare body in the presence of a dense one, as in the explosion of a gun, or by the return to their natural dispositions of bodies that have been distorted or bent, as manifested by the action of a bow.

Turning from the passages on “force” to those on “impetus,” it is immediately apparent that Leonardo has absorbed the medieval theory that explains the motion of projectiles by the impression of an impetus into the projectile by the projector, a theory outlined in its most mature form by Jean Buridan and repeated by many other authors, including Albert of Saxony, whose works Leonardo had read.24 Leonardo’s dependence on the medieval impetus theory is readily shown by noting a few of his statements concerning it:

1. Ms E, folio 22r: “Impetus is a virtue [ “virtu” ] created by motion and transmitted by the motor to the mobile that has as much motion as the impetus has life.”

2. Codex Atlanticus, folio 161v-a: “Impetus is a power [ “potenzia” ] of the motor applied to its mobile, which [power] causes the mobile to move after it has separated from its motor.”

3. MS G, folio 73r: “Impetus is the impression of motion transmitted by the motor to the mobile. Impetus is a power impressed by the motor in the mobile…. Every impression tends toward permanence or desires to be permanent.”

In the last passage Leonardo’s words are particularly reminiscent of Buridan’s quantitative description of impetus as directly proportional to both the quantity of prime matter in and the velocity of the mobile is nowhere evident in Leonardo’s notebooks. In some passages Leonardo noted the view held by some of his contemporaries (such as Agostino Nifo) that the air plays a supplementary role in keeping the projectile in motion (see Codex Atlanticus, fol. 168v-b): “Impetus i s[the] impression of local motion transmitted by the motor to the mobile and maintained by the air or the water as they move in order to prevent a vacuum” (cf.ibid., fol. 219v-a). In Codex Atlanticus, folio 108r-a, however, he stressed the role of the air in resisting the motion of the projectile and concluded that the air gives little or no help to the motion. Furthermore, in Codex Leicester, folio 29v, he gives a long and detailed refutation of the possible role of the air as motor, as Buridan had before him. And not only is it the air as resistance that weakens the impetus in a projectile; the impetus is also weakened and destroyed by the tendency to natural motion. For example, in MS E, folio 29r, Leonardo says: “But the natural motion conjoined with the motion of a motor [that is, arising from the impetus derived from the motor] consumes the impetus of the motor.”

Leonardo also applied the impetus theory to many of the same inertial phenomena as did his medieval predecessors—for instance, to the stability of the spinning top (MS E, fol. 50v), to pendular and other kinds of oscillating motion (Codex Arundel, fol. 2r), to impact and rebound (Codex Leicester, fols. 8r, 29r), and to the common medieval speculation regarding a ball falling through a hole in the earth to the center and rising on the other side before falling back and oscillating about the center of the earth (Codex Atlanticus, fol. 153v-b). In a sense, all of these last are embraced by the general statement in MS E, folio 40v: “The impetus created in whatever line has the power of finishing in any other line.”

In the overwhelming majority of passages, Leonardo applied the impetus theory to violent motion of projection. In some places, however, he also applied it to cases of natural motion, as did his medieval predecessors when they explained the acceleration of falling bodies through the continuous impression of impetus by the undiminished natural weight of the body. For example, in Codex Atlanticus, folio 176r-a, Leonardo wrote: “Impetus arises from weight just as it arises from force.” And in the same manuscript (fol. 202v-b) he noted the continuous acquisition of impetus “up to the center [of the world].” Leonardo was convinced of the acceleration of falling bodies (although his kinematic description of that fall was confused); hence, he no doubt believed that the principal cause of such acceleration was the continual acquisition of impetus.

One last aspect of Leonardo’s impetus doctrine remains to be discussed-his concept of compound impetus, defined in MS E, folio 35r, as that which occurs when the motion “partakes of the impetus of the motor and the impetus of the mobile.” The example he gave is that of a spinning body moved by an external force along a straight line. When the impetus of the primary force dominates, the body moves along a simple straight line. As that impetus dies, the rotary motion of the spinning body acts with it to produce a composite-curved motion. Finally, all of the impetus of the original motion is dissipated and a simple circular motion remains that arises only from the spinning body. A series of passages (Codex Arundel, fols. 143r-144v) are further concerned with the relationship of transversal and natural motions, which, if the passages and diagrams have been understood correctly, concur to produce resultant composite motions. It appears later in the same manuscript (fol. 147v) that Leonardo thought that the first part of a projectile path was straight until the primary impetus diminished enough for the natural motion to have an effect:

The mobile is [first] moved in that direction [ “aspetto” ] in which the motion of its motor is moved. The straightness of the transversal motion in the mobile lasts so long as the internal power given it by the motor lasts. Straightness is wanting to the transversal motion because [that is, when] the power which the mobile acquired from its motor diminishes.

A beautiful instance of compound motion upon which Leonardo reported more than once is that of an arrow or stone shot into the air from a rotating earth, which arrow or stone would fall to the ground with rectilinear motion with respect to the rotating earth because it receives a circular impetus from the earth. But with respect to a stationary frame, the descent is said to be spiral, that is, compounded of

rectilinear and circular motions. The longer of the passages in which Leonardo described this kind of compound motion is worth quoting (MS G, fol. 55r; see Figure 19):

On the heavy body descending in air, with the elements rotating in a complete rotation in twenty-four hours. The mobile descending from the uppermost part of the sphere of fire will produce a straight motion down to the earth even if the elements are in a continuous motion of rotation about the center of the world. Proof : let b be the heavy body which descends through the elements from [point] a to the center of the world m. I say that such a heavy body, even if it makes a curved descent in the manner of a spiral line, will never deviate in its rectilinear descent along which it continually proceeds from the place whence it began to the center of the world, because when it departs from point a and descends to b, in the time in which it has descended to b, it has been carried on to [point] d, the position of a having rotated to c, and so the mobile finds itself in the straight line extending from c to the center of the world, m. If the mobile descends from d to f, the beginning of motion, c, is, in the same time, moved from c to f [le]. And if f descends to h e is rotated to g; and so in twenty-four hours the mobile descends to the earth [directly] under the place whence it first began. And such a motion is a compounded one.

[In margin:] If the mobile descends from the uppermost part of the elements to the lowest point in twenty-four hours, its motion is compounded of straight and curved [motions]. I say “straight” because it will never deviate from the shortest line extending from the place whence it began to the center of the elements, and it will stop at the lowest extremity of such a rectitude, which stands, as if to the zenith, under the place from which the mobile began [to descend]. And such a motion is inherently curved along with the parts of the line, and consequently in the end is curved along with the whole line. Thus it happened that the stone thrown from the tower does not hit the side of the tower before hitting the ground.

This is not unlike a passage found in Nicole Oresme’s Livre du ciel et du monde.25

In view of the Aristotelian and Scholastic doctrines already noted, it is not surprising to find that Leonardo again and again adopted some form of the Peripatetic law of motion relating velocity directly to force and inversely to resistance. For example, MS F, folios 51r-v, lists a series of Aristotelian rules. It begins “1° if one power moves a body through a certain space in a certain time, the same power will move half the body twice the space in the same time….” (compare MS F, fol. 26r). This law, when applied to machines like the pulley and the lever, became a kind of primitive conservation-of-work principle, as it had been in antiquity and the Middle Ages. Its sense was that “there is in effect a definite limit to the results of a given effort, and this effort is not alone a question of the magnitude of the force but also of the distance, in any given time, through which it acts. If the one be increased, it can only be at the expense of the other.” 26 In regard to the lever, Leonardo wrote (MS A, fol. 45r): “The ratio which the length of the lever has to the counterlever you will find to be the same as that in the quality of their weights and similarly in the slowness of movement and in the quality of the paths traversed by their extremities, when they have arrived at the permanent height of their pole.” He stated again (E, fol. 58v):

By the amount that accidental weight is added to the motor placed at the extremity of the lever so does the mobile placed at the extremity of the counterlever exceed its natural weight. And the movement of the motor is greater than that of the mobile by as much as the accidental weight of the motor exceeds its natural weight.

Leonardo also applied the principle to more complex machines (MS A, fol. 33v): “The more a force is extended from wheel to wheel, from lever to lever, or from screw to screw, the greater is its power and its slowness.” Concerning multiple pulleys, he added (MS E, fol. 20v):

The powers that the cords interposed between the pulleys receive from their motor are in the same ratio as the speeds of their motions. Of the motions made by the cords on their pulleys, the motion of the last cord is in the same ratio to the first as that of the number of cords; that is, if there are five, the first is moved one braccio, while the last is moved 1/5 of a braccio; and if there are six, the last cord will be moved 1/6 of a braccio, and so on to infinity. The ratio which the motion of the motor of the pulleys has to the motion of the weight lifted by the pulleys is the same as that of the weight lifted by such pulleys to the weight of its motor….

It is not difficult to see why Leonardo, so concerned with this view of compensating gain and loss, attacked the speculators on perpetual motion (Codex Forster II2, fol. 92v): “O speculators on continuous motion, how many vain designs of a similar nature have you created. Go and accompany the seekers after gold.”

One area of dynamics that Leonardo treated is particularly worthy of note, that which he often called “percussion.” In this area he went beyond his predecessors and, one might say, virtually created it as a branch of mechanics. For him the subject included not only effects of the impacts of hammers on nails and other surfaces (as in MS C, fol. 6v; MS a, fol. 53v) but also rectilinear impact of two balls, either both in motion or one in motion and the other at rest (see the various examples illustrated on MS A, fol. 8r), and rebound phenomena off a firm surface. In describing impacts in Codes Arundel,folio 83v, Leonardo wrote: “There are two kinds [ “nature” ] of percussion: the one when the object [struck] flees from the mobile that strikes it; the other when the mobile rebounds rectilinearly from the object struck.” In one passage (MS I, fol. 41v), a problem of impacting balls is posed: “Ball a is moved with three degrees of velocity and ball b with four degrees of velocity. It is asked what is the difference [“variety”] in such percussion [of a] with b when the latter ball would be at rest and when it [a] would meet the latter ball [moving] with the said four degrees of velocity.”

In some passages Leonardo attempted to distinguish and measure the relative effects of the impetus of an object striking a surface and of the percussion executed by a resisting surface. For example, in Codex Arundel, folio 81v, he showed the rebound path as an arc (later called “l’arco del moto refresso”) and indicated that the altitude of rebound is acquired only from the simple percussion, while the horizontal distance traversed in rebound is acquired only from the impetus that the mobile had on striking the surface, so that “by the amount that the rebound is higher than it is long, the power of the percussion exceeds the power of the impetus, and by the amount that the rebound’s length exceeds its height, the percussion is exceeded by the impetus.”

What is perhaps Leonardo’s most interesting conclusion about rebound is that the angle of incidence is equal to that of rebound. For example, in Codex Arundel, folio 82v, he stated: “The angle made by the reflected motion of heavy bodies becomes equal to the angle made by the incident motion.” Again, in MS A, folio 19r (Figure 20), he wrote:

Every blow struck on an object rebounds rectilinearly at an angle equal [ “simile” ] to that of percussion. This proposition is clearly evident, inasmuch as, if you would strike a wall with a ball, it would rise rectilinearly at an angle equal to that of the percussion. That is, if

the ball b is thrown at c, it will return rectilinearly through the line eb because it is constrained to produce equal angles on the wall fg. And if you throw it along line bd, it will return rectilinearly along line de, and so the line of percussion and the line of rebound will make one angle on wall fg situated in the middle between two equal angles, as d appears between in and n. [See also Codex Atlanticus, fol. 125r-a.]

The transfer, and in a sense the conservation, of power and impetus in percussion is described in Codex Leicester, folio 8r:

If the percussor will be equal and similar to the percussed, the percussor leaves its power completely in the percussed, which flees with fury from the site of the percussion, leaving its percussor there. But if the percussor-similar but not equal to the percussed—is greater, it will not lose its impetus completely after the percussion but there will remain the amount by which it exceeds the quantity of the percussed. And if the percussor will be less than the percussed, it will rebound rectilinearly through more distance than the percussed by the amount that the percussed exceeds the percussor.

Leonardo is here obviously groping for adequate laws of impact.

The last area to be considered is Leonardo’s treatment of the kinematics of moving bodies, especially the kinematics of falling bodies. In Codex Arundel, folio 176v, he gave definitions of “slower” and “quicker” that rest ultimately on Aristotle:27 “That motion is slower which, in the same time, acquires less space. And that is quicker which, in the same time, acquires more space.” The description of falling bodies in respect to uniform acceleration is, of course, more complex. It should be said at the outset that Leonardo never succeeded in freeing his descriptions from essential confusions of the relationships of the variables involved. Most of his passages imply that the speed of fall is not only directly proportional to the time of fall, which is correct, but that it is also directly proportional to the distance of fall, which is not. In MS M, folio 45r, lie declared that “the gravity [that is, heavy body] which descends freely, in every degree of time, acquires a degree of motion, and, in every degree of motion, a degree of speed.” If, like Duhem, one interprets “degree of motion,” that is, quantity of motion, to be equivalent not to distance but to the medieval impetus, then Leonardo’s statement is entirely correct and implies only that speed of fall is proportional to time of fall. One might also interpret the passage in MS M, folio 44r, in the same way (see Figure 21):

Prove the ratio of the time and the motion together with the speed produced in the descent of heavy bodies

by means of the pyramidal figure, for the aforesaid powers [ “potenzie” ] are all pyramidal since they commence in nothing and go on increasing by degrees in arithmetic proportion. If you cut the pyramid in any degree of its height by a line parallel to the base, you will find that the space which extends from the section to the base has the same ratio as the breadth of the section has to the breadth of the whole base. You see that [just as] ab is 1/4 of ae so section fb is 1/4 of ne.

As in mathematical passages, Leonardo here used “pyramidal” where “triangular” is intended.28 Thus he seems to require the representation of the whole motion by a triangle with point a the beginning of the motion and ne the final speed, with each of all the parallels representing the speed at some and every instant of time. In other passages Leonardo clearly coordinated instants in time with points and the whole time with a line (see Codex Arundel, fols. 176r-v). His triangular representation of quantity of motion is reminiscent of similar representations of uniformly difform motion (that is, uniform acceleration) in the medieval doctrine of configurations developed by Nicole Oresme and Giovanni Casali ; Leonardo;s use of an isosceles triangle seems to indicate that it was Casali’s account rather than Oresme’s that influenced him.29 It should be emphasized, however, that in was applying the triangle specifically to the motion of fall rather than to an abstract example of uniform acceleration, Leonardo was one step closer to the fruitful use to which Galileo and his successors put the triangle. two similar passages illustrate this-the first (MS M, fol. 59v) again designates the triangle as a pyramidal figure, while in the second, in Codex Madrid I, folio 88v (Figure 22)30 the triangle is divided into sixteen equally spaced sections. Leonardo explained the units on the left of the latter figure by saying that “these unities are designated to demonstrate that the excesses of degrees are equal.” Lower on the same page he noted that “the thing which descends acquires a degree of speed in every degree of motion and loses a degree of time.” By “every degree of motion” he may have meant equal vertical spaces between the parallels into which the motion is divided. Hence, this phrase would be equivalent to saying “in every degree of time.” The comment about the loss of time merely emphasizes the whole time spent during the completion of the motion.

All of the foregoing comments suggest a possible, even plausible, interpretation of Leonardo’s concept of “degree of motion” in these passages. Still, one should examine other passages in which Leonardo seems also to hold that velocity is directly proportional to distance of fall. Consider, for example, MS M, folio 44v: “The heavy body [“gravita”] which descends, in every degree of time, acquires a degree of motion more than in the degree of time preceding, and similarly a degree of speed [“velocity”] more than

in the preceding degree of motion. Therefore, in each doubled quantity of time, the length of descent [ “lunghezza del discenso” ] is doubled and [also] the speed of motion.” The figure accompanying this passage (Figure 23) has the following legend: “It is here shown that whatever the ratio that one quantity of time has to another, so one quantity of motion will have to the other and [similarly] one quantity of speed to the other.” There seems little doubt from this passage that Leonardo believed that in equal periods of time, equal increments of space are being acquired. One last passage deserves mention because, although also ambiguous, it reveals that Leonardo believed that the same kinds of relationships hold for motion on an incline as in vertical fall (MS M,fol. 42v): “Although the motion be oblique, it observes, in its every degree, the increase of the motion and the speed in arithmetic proportion.” The figure (Figure 24) indicates that the motion on the incline is represented by the triangular section ebc, while the vertical fall is represented by abc, Hence, with this figure Leonardo clearly intended that the velocities at the end of both vertical and oblique descents are equal (that is, both are represented by bc) and also that the velocities midway in these descents are

equal (that is, inn = op). The figure also shows that the times involved in acquiring the velocities differ, since the altitude of ∆ebc is obviously greater than the altitude of ∆abc.

So much, then, for the most important aspects of Leonardo’s theoretical mechanics. His considerable dependence on earlier currents has been noted, as has his quite significant original extension and development of those currents. It cannot be denied, however, that his notebooks, virtually closed as they were to his successors, exerted little or no influence on the development of mechanics.

NOTES

1. The many passages from Leonardo’s notebooks quoted here can be found in the standard eds. of the various MSS . Most of them have also been collected in Uccelli’s ed. of I libri di neccanica. The English trans. (with only two exceptions) are my own.

2. The full corpus has been published in E. A. Moody and M. Clagett, The Medieval Science of Weights (Madison, Wis., 1952; repr., 1960). Variant versions of the texts have been studied and partially published in J. E. Brown, “The ’Scientia de ponderibus’ in the Later Middle Ages” (disseration, Univ. of Wis., 1967).

3. Moody and Clagett, op. cit., pp. 128-131. Incidentally, the definition that precedes the proof is the sure sign that Leonardo translated the postulates and first two enunciations from the Elementa rather than from version P (where the enunciations are the same), since the definition was not included in Version P .

4.Ibid., pp. 174–207.

5.Ibid., pp. 130–131.

6.Ibid., pp. 188–191. Leonardo’s translation of these same propositions in Codex Atlanticus, fols. 154v-a-r-a, is very literal: “La equalita della declinazione conserve la equalita del peso. Se per due vie di diverse obliquity due pesi discendano, e sieno medesimo proporzione, se della d[eclinazione] de’ pesi col medesimo ordine presa sara ancora una medesima virtb dell’una e d[ell’altra in discendendo].” The bracketed material has been added from the Latin text.

7. Marcolongo, Studi vinciani, p. 173.

8. Duhem, Les origines de to statique, I , 189-190.

9. Moody and Clagett, op. cit., pp. 184-187. Leonardo’s more literal translation of R1.08 in Codex Atlanticus, fol. 154v-a, runs: “Se le braccia della libra sono inequali e net centro del moto faranno un angolo, s’e’ termini loro s’accosteranno parte equalmente alla direzione, e’ pesi equali in questa disposizione equalmente peseranno.”

10. Marcolongo, op. cit., p. 149, discusses one such passage (Codex Arundel, fol. 67v); see also pp. 147–148, discussing the figures on MS Ashburnham 2038, fol. 3r; and Codex Arundel, fol. 32v (in the passage earlier than the one noted above in the text).

11. Moody and Clagett, op. cit., pp. 208–211.

12.Ibid., pp. 210–211.

13.Ibid., pp. 64–65, 102–109, 192–193.

14. Clagett, “Leonardo da Vinci …,” pp. 119–140.

15.Ibid., pp. 121–126.

16.Ibid., p. 126.

17. Archimedes, Momunenta omnia mathenmtica, quae extantex traditione Francisci Maurolici (Palermo, 1685), De momentis aequalibus, bk. IV, prop. 16, pp. 169-170. Maurolico completed the De momentis aequalibus in 1548.

18. Marcolongo, op. cit., pp. 230–216.

19. See Clagett, “Leonardo da Vinci …,” pp. 140–141. That account is here revised, taking into account the probability that the fragments were not copied by Leonardo.

20.Ibid., p. 140, n. 65, notes the opinion of Favaro and Schmidt that the fragments are truly in Leonardo’s hand. But Carlo Pedretti, whose knowledge of leonardo’s hand is sure and experienced, is convinced they are not. Arredi, Le origini dell’idrostatica, pp. 11–12, had already recognized that the notes were not in Leonardo’s hand.

21. M. Clagett, The Science of Mechanics in the Middle Ages (Madison, Wis., 1959; repr., 1961), pp. 124–125.

22. Here Arredi’s account is followed closely.

23. Clagett, The Science of Mechanics, p. 187.

24.Ibid., chs. 8–9.

25.Ibid., pp. 601–603.

26. Hart, The Mechanical Investigations, pp. 93–94.

27. Clagett, The Science of Mechanics, pp. 176–179.

28. Clagett, “Leonardo da Vinci …,” p. 106, quoting MS K , fol. 79v.

29. M. Clagett, Nicole Oresme and the Medieval Geometry of Qualities (Madison, Wis., 1968), pp. 66–70.

30. I must thank L. Reti, editor of the forthcoming ed. of the Madrid codices, for providing me with this passage.

Marshall Clagett

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mechanics

mechanics, branch of physics concerned with motion and the forces that tend to cause it; it includes study of the mechanical properties of matter, such as density, elasticity, and viscosity. Mechanics may be roughly divided into statics and dynamics; statics deals with bodies at rest and is concerned with such topics as buoyancy, equilibrium, and the principles of simple machines, while dynamics deals with bodies in motion and is sometimes further divided into kinematics (description of motion without regard to its cause) and kinetics (explanation of changes in motion as a result of forces). A recent subdiscipline of dynamics is nonlinear dynamics, the study of systems in which small changes in a variable may have large effects. The science of mechanics may also be broken down, according to the state of matter being studied, into solid mechanics and fluid mechanics. The latter, the mechanics of liquids and gases, includes hydrostatics, hydrodynamics, pneumatics, aerodynamics, and other fields.

Early Mechanics

Mechanics was studied by a number of ancient Greek scientists, most notably Aristotle, whose ideas dominated the subject until the late Middle Ages, and Archimedes, who made several contributions and whose approach was quite modern compared to other ancient scientists. In the Aristotelian view, ordinary motion required a material medium; a body was kept in motion by the medium rushing in behind it in order to prevent a vacuum, which, according to this philosophy, could not occur in nature. Celestial bodies, on the other hand, were kept in motion through the vacuum of space by various agents that, in the Christianized version of Aquinas and others, acquired an angelic character.

This explanation was rejected in the 14th cent. by several philosophers, who revived the impetus theory proposed by John Philoponos in the 6th cent. AD; according to this theory a body acquired a quantity called impetus when it was set in motion, and it eventually came to rest as the impetus died out. The impetus school flourished in Paris and elsewhere during the 14th and 15th cent. and included William of Occam (Ockham), Jean Buridan, Albert of Saxony, Nicolas Oresme, and Nicolas of Cusa, although it was never successful in replacing the dominant Aristotelian mechanics.

Modern Mechanics

Modern mechanics dates from the work of Galileo, Simon Stevin, and others in the late 16th and early 17th cent. By means of experiment and mathematical analysis, Galileo made a number of important studies, particularly of falling bodies and projectiles. He enunciated the principle of inertia and used it to explain not only the mechanics of bodies on the earth but also that of celestial bodies (which, however, he believed moved in uniform circular orbits). The philosopher René Descartes advocated the application of the mathematical-mechanical approach to all fields and founded the mechanistic philosophy that was so important in science for the next two centuries or more.

The first system of modern mechanics to explain successfully all mechanical phenomena, both terrestrial and celestial, was that of Isaac Newton, who in his Principia (Mathematical Principles of Natural Philosophy, 1687) derived three laws of motion and showed how the principle of universal gravitation can be used to explain both the behavior of falling bodies on the earth and the orbits of the planets in the heavens. Newton's system of mechanics was developed extensively over the next two centuries by many scientists, including Johann and Daniel Bernoulli, Leonhard Euler, J. le Rond d'Alembert, J. L. Lagrange, P. S. Laplace, S. D. Poisson, and W. R. Hamilton. It found application to the explanation of the behavior of gases and thermodynamics in the statistical mechanics of J. C. Maxwell, Ludwig Boltzmann, and J. W. Gibbs.

In 1905, Albert Einstein showed that Newton's mechanics was an approximation, valid for cases involving speeds much less than the speed of light; for very great speeds the relativistic mechanics of his theory of relativity was required. Einstein showed further in his general theory of relativity (1916) that gravitation could be explained in terms of the effect of a massive body on the framework of space and time around it, this effect applying not only to the motions of other bodies possessing mass but also to light. In the quantum mechanics developed during the 1920s as part of the quantum theory, the motions of very tiny particles, such as the electrons in an atom, were explained using the fact that both matter and energy have a dual nature—sometimes behaving like particles and other times behaving like waves. Two different but mathematically equivalent forms of quantum mechanics were elaborated, the wave mechanics of Erwin Schrödinger and the matrix mechanics of Werner Heisenberg.

Bibliography

See I. B. Cohen, Introduction to Newton's Principia (1971); E. Mach, Science of Mechanics (6th ed. 1973); J. Gleick, Chaos (1987).

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mechanics

mechanics Branch of physics concerned with the behaviour of matter under the influence of forces. It may be divided into solid mechanics and fluid mechanics. Another classification is as statics – the study of matter at rest – and dynamics – the study of matter in motion. In statics, the forces on an object are balanced and the object is said to be in equilibrium; static equilibrium may be stable, unstable or neutral. Dynamics may be further divided into kinematics – the description of motion without regard to cause – and kinetics – the study of motion and force. Classical dynamics rests primarily on Isaac Newton's laws of motion. Modern physics has shown these laws to be special cases similar to more general laws. Relativistic mechanics deals with the behaviour of matter at high speeds, approaching that of light, whereas quantum mechanics deals with the behaviour of matter at the level of atoms and molecules. See also quantum theory; relativity

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mechanics

me·chan·ics / məˈkaniks/ • pl. n. 1. [treated as sing.] the branch of applied mathematics dealing with motion and forces producing motion. ∎  machinery as a subject; engineering. 2. the machinery or working parts of something: he looks at the mechanics of a car before the bodywork. ∎  the way in which something is done or operated; the practicalities or details of something: the mechanics of cello playing.

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